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Thursday, July 24, 2014

One Resource a (Week)Day #17: Interactive Tasks from NRich

I have been doing some more playing of the secondary-school tasks from NRich, and I noticed in that process that they actually have some really nice interactive applets. I think that making an effective teaching applet is tricky, because:

1. If you make an applet that has too many features, even if you have the best of intentions, it can end up distracting from the actual mathematics.

2. If you make an applet that has too few features, on the other hand, it does not necessarily support the student's need to generate more data points and to test their conjectures.

Anyhow, here are a few tasks that have quite nice connections to high-school topics, each with a useful interactive applet.

http://nrich.maths.org/2293 has to do with finding (and predicting) areas of tilted squares, with a specified tilt k. The problem is accessible with just basic geometry, but it is extendable to a function of two input variables. You can generalize the pattern A(t, h) to describe the area of a square with a tilt t and whose two leftmost vertices differ by h units in height. The applet at the bottom of that page is very user-friendly. It only has two togglable points for you to construct squares of a certain tilt and height, and it is only there to help students construct newer instances and to observe their resulting shape and area concretely.

http://nrich.maths.org/2281 is a super easy-entry puzzle on building a pyramid of numbers. The guiding questions are gentle but they effectively get the kids to start thinking about how the position of a bottom number affects the final value at the top of the pyramid. They can make conjectures and test them repeatedly using the applet, thereby deepening their observations along the way. And then, the plot thickens when the pyramid gets to be bigger -- with 4 or more elements at the bottom level. Eventually, it could be generalized to show connections to Pascal's Triangle, a topic often touched upon in Algebra 2. Tres cool!

http://nrich.maths.org/7016 is a quite high-level task suitable for thinking about sequences. The applet is there for the students to try and gather data about which numbers will light up each color, and the really nice thing is that each group can be working on different patterns, without extra work on your part to generate different data. The entry to this task is a fairly straight-forward practice of linear equations / sequences, but when you start asking questions about how to light up multiple colors, the question gets rich really fast. When we dig even deeper into how to generalize relationships between sequences, I at least found myself in a quick sand. Besides some trial-and-error, I couldn't find a systematic way of predicting the first sequence element where two lights (of known pattern) will both light up. (After the first coinciding lighting, the rest is easy to obtain.) Can you help?

By the way, I am loving the various Twitter quotes from the Twitter Math Camp y'all are at. Keep them coming! You guys are so inspiring!

Addendum 7/24/14: I did a bit of playing and figured out how to find the first coinciding element of two linear sequences! For example, this problem from Amy Gruen goes nicely with the problem #7016 from above (hits the same type of idea). I leave it for you as an exercise to find all the numbers that satisfy this within the range 1 to 3000, but I'll answer it in a few days if you haven't already figured out how to do it...

I am a number less than 3000, Divide by 32, the remainder is 30. Divide by 58, the remainder is 44. Who am I?